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Given a line with slope $m$, if the line is rotated $90^{\circ}$ clockwise the slope of the resulting figure is $-m$, how can I determine the slope if the line is rotated $30^{\circ}$ or $60^{\circ}$ clockwise.

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For a $90^{\circ}$ rotation, the slope becomes $-\frac{1}{m}$, not $-m$. For a smaller rotation, you can take the arctangent of $m$, add $30^{\circ}$ or $60^{\circ}$ and take the tangent, or use the angle addition formulas with tangent $30^{\circ}$ or $60^{\circ}$

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